Ordered Size Ramsey Number of Paths
J\'ozsef Balogh, Felix Christian Clemen, Emily Heath, Mikhail Lavrov

TL;DR
This paper investigates the ordered size Ramsey number for paths, establishing bounds that relate the number to the path lengths and extending recent results in oriented graph Ramsey theory.
Contribution
It provides new bounds on the ordered size Ramsey number of paths, connecting it to path lengths and logarithmic factors, and extends recent oriented graph results.
Findings
Lower bound: r^2 s
Upper bound: C r^2 s ( log s)^3
Extends results from oriented graph Ramsey theory
Abstract
An ordered graph is a simple graph with an ordering on its vertices. Define the ordered path to be the monotone increasing path with edges. The ordered size Ramsey number is the minimum number for which there exists an ordered graph with edges such that every two-coloring of the edges of contains a red copy of or a blue copy of . For , we show , where is an absolute constant. This problem is motivated by the recent results of Buci\'c-Letzter-Sudakov and Letzter-Sudakov for oriented graphs.
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